Assignment 2: Relativity of Simultaneity

For Einstein, the big breakthrough in his work on special relativity came
when he found a way to reconcile the principle of relativity and the light
postulate. He recognized that these principles only seemed irreconcilable
because of an unwarranted assumption that we routinely make about space and
time. We assume that all observers should agree on which events are
simultaneous. Instead, Einstein noticed, we may allow for the possibility that
observers in relative motion may disagree about which spatially separated
events are simultaneous. This assumption of the relativity of simultaneity
allowed him to retain both the principle of relativity and the light postulate.
This assignment will help you to see how.

1. An observer is at the midpoint of a long
spaceship. At the same instant he sends light signals to both front and rear of
the spaceship. Event A is the arrival of the signal at the rear; event B is the
arrival of the signal at the front.

(a) Are the two events A and B simultaneous according to the spaceship
observer?

(b) Imagine that there are two good clocks located at the front and the rear
of the spaceship and the arrival of the signals is used to reset each clock to
the same time. Are the clocks now properly synchronized according to the
spaceship observer?

(c) The spaceship is moving rapidly in the direction of its length past a
planet. An observer on the planet watches the signaling procedure described
above. Does the planet observer judge events A and B to be simultaneous? If
not, which happens first?

(d) Does the planet observer judge the two clocks to be set in proper
synchrony? If not, which is set ahead of the other?

2. A light signal flashes back and forth between
the two ends of the same spaceship. If the light postulate is to hold for the
spaceship observer, then the spaceship observer must judge that the light
travels at the same speed in all directions. That is, according to the
spaceship observer, the signal must take the same time to travel from front to
back as from back to front. Assume this transit time is one minute. Then the
arrival times of the light signal must be registered as 12:00, 12:02, 12:04,
...etc. at the rear of the ship and 12:01, 12:03, 12:05, ... etc. at the
front.

(a) Assume the light postulate also holds for the planet observer. Will the
planet observer judge the transit time for the forward trip of the light signal
to be the same as the transit time for the backward trip? If not, which is
longer?

(b) How can the planet observer reconcile the answer to 2.(a) with the
readings on the clocks of the moving spaceship that record the transit times
for the light signal?

For discussion in the recitation.

A. Two observers I and II both stand on a large
platform. There are two lightning strikes, A and B.

The observer I is located at the midpoint of the spatial locations of the
strikes A and B. Light signals coming from the strikes A and B arrive at this
observer I at the same time.

The observer II is located much closer to the strike A. As a result, the
light signal from strike A arrives at observer II much earlier than the light
signal from strike B.

Observer I sees the signals at the same time; observer II sees them at
different times.

Is this difference the relativity of simultaneity of relativity theory? If
not, why not?

B. Two identical spaceships pass one another,
moving rapidly in opposite directions at the same speed according to an
observer on a nearby planet. The planet observer judges that both spaceships
have shrunk the same amount due to relativistic length contraction. So they are
the same length and, in conformity with this expectation, the planet observer
notes that the two spaceships line up perfectly as they pass.

An observer on one of the spaceships, however, finds the other spaceship to
be moving rapidly. So that spaceship observer judges the other spaceship to
have shrunk relative to the first spaceship. And an observer on the second
spaceship comes to the reverse judgment, that the first spaceship has
shrunk.

How is it possible for all of of them to come to such different
judgements?

C. Judgments of simultaneity are involved in any
procedure that measures the length of moving bodies or the times elapsed for
processes on them. Consider some procedures for measuring such lengths and
times and show how judgments of simultaneity are hidden in them. What if, for example we measure the length of a moving
body by timing how long it takes to pass a single observation point, where we
use just one clock to time its passage. Its length is just its speed multiplied
by the time measured.